Powers of sequences and recurrence

Nikos Frantzikinakis (1) , Emmanuel Lesigne (2) , Maté Wierdl (1)

1 Department of Mathematics, University of Crete
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mendès-France, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.

Domaines: Systèmes dynamiques [math.DS]

Nikos Frantzikinakis, Emmanuel Lesigne, Maté Wierdl. Powers of sequences and recurrence. Proceedings of the London Mathematical Society, 2009, 98 (2), pp.504-530. ⟨hal-00276737⟩